Also for watering: the soil looks too dry too soon currently. Mine, too--and I have no real problems. The very top surface of the soil dries out really fast in sun and wind, but that doesn't mean there isn't water underneath. You can test that by pushing a screwdriver into the soil. If it goes in fairly easily (barring rocks), your soil is damp. Or drive a small trowel into it and separate it into a V shape by rocking it a little bit. That'll give you a direct visual on how deep the water is. Here would be my plan: for now do very frequent waterings to help really wet the soil, I'm not familiar with California (except for the part about moderate temperatures and no rain), but this doesn't sound like a good idea--it encourages shallow roots and lawn diseases. The standard is 1" of water a week, all at once (well, within a day to allow for on/off cycles to allow the water to penetrate). For sandy soil your mileage will differ because the water wicks out fast and goes very deep very quickly. I will also probably try milorganite as a jump start, as I fear my soil is probably pretty depleted. I swear by it, others don't (it is processed sewage, but heated enough to kill everything in it). The 18 lbs/thousand square feet I put down last weekend kicked in and it's a rainy week. The grass, since Sunday, has grown one and a half inches and gone a deeper green. Once it stops raining, I'll mow. I went high, but I'm going to renovate this fall--with a deep, heavy core aeration I want lots of organics mixing into the holes. I alternate it with alfalfa and soy. When I started (early last year), my soil profile was slightly silty clay, poor drainage, very poor water penetration, hard as a rock, and a light tan. It had been corn field for as long as I could remember prior to building, and even my parents don't remember when it wasn't. Call it 40 years of abuse. Now it's still slightly silty clay (nothing's going to change that short of a renovation down to a foot or more). Drainage is fair to good, penetration is good to excellent, the soil is still hard when dry (it's supposed to be), but spongy when wet. The top half-inch is black, another inch is deep brown, and a good three inches below that have turned a midrange brown. Everything grows better and more strongly. I'm very pleased. Some older photos are here. I have to update them--the grass has improved since these were taken and the season advanced a bit.

Not sure what the mud test is but what you're doing is great. Search this forum for leveling.

What about compost? If you haven't done so, 1 cubic yard per 1000 sqft will help a lot. Your soil may lack sufficient soil microbes (or at least diversity of species) to take advantage of SBM and Alfalfa. You can look at soil biology as a microscopic world with billions of people with ALL KINDS OF JOBS to keep the world spinning... Chemical products might have killed off some of the important jobs... Only compost can provide those "jobs"...

The information you get from the "mud" or "jar" test is about the amount of organic matter in the soil, the stuff that floats on top of everything else since very few that do make the effort will have the idea soil, loam. Pouring "fertilizers" on a lawn simple creates more problems because that stimulates lush green, and sick, plant growth that then needs other poisons to control. "Conventional" lawn care is a vicious cycle of spending lots of money to have a sick lawn. Work at getting the necessary organic matter into the soil and have a good reliable soil test done. Here is a link that might be useful: OSU CES